The Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution, is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
Let 
represent the angle that a line, with fixed point of rotation,
makes with the vertical axis, as shown above. Then
so the distribution of angle 
is given by
 |
(5) |
This is normalized over all angles, since
 |
(6) |
and
The general Cauchy distribution and its cumulative distribution can be written as
where 
is the half width at half maximum and 
is the statistical median.
In the illustration about, 
.
The Cauchy distribution is implemented in the Wolfram Language as CauchyDistribution[m, Gamma/2].
The characteristic function is
The moments 
of the distribution are undefined since the integrals
 |
(14) |
diverge for 
.
If 
and 
are variates with a normal distribution, then
has a Cauchy distribution with statistical median 
and full width
 |
(15) |
The sum of 
variates each from a Cauchy distribution has itself a Cauchy distribution, as can
be seen from
where 
is the characteristic function and ](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FCauchyDistribution%2FInline53.svg){kind=small}
is the inverse Fourier
transform, taken with parameters 
.
See also
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References
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 104, 1984.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 114-115, 1992.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyDistribution.html